課程資訊
課程名稱
代數導論一
Introduction to Algebra (Ⅰ) 
開課學期
101-1 
授課對象
電機資訊學院  電機工程學系  
授課教師
王姿月 
課號
MATH2105 
課程識別碼
201 24210 
班次
 
學分
全/半年
半年 
必/選修
選修 
上課時間
星期一3,4(10:20~12:10)星期四7,8(14:20~16:20) 
上課地點
新204新204 
備註
總人數上限:100人
外系人數限制:10人 
Ceiba 課程網頁
http://ceiba.ntu.edu.tw/1011abstactalgebraW 
課程簡介影片
 
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課程概述

*Contents:
1. Integers and Permutations: divisors and prime factorization, integers modulo n, permutations
2. Groups: groups, subgroups, cyclic groups, homomorphisms, cosets, Lagrange theorem, groups of motions and symmetries, isomorphism theorem
3. Rings: examples and basic properties, integral domains and fields, ideal and factor rings, homomorphisms, polynomial rings, symmetric polynomials, unique factorization domains, principal ideal domain

4. Fields: algebraic extensions, splitting fields, finite fields, geometric construction(ruler and compass), fundamental theorem of algebra
 

課程目標
Understand the structure of groups, rings, and fields. 
課程要求
 
預期每週課後學習時數
 
Office Hours
每週五 15:00~16:00
每週一 15:00~16:00 
參考書目
T. W. Judson, Abstract Algebra Theory and Applications
(can be obtained via http://abstract.pugetsound.edu/download.html
 
指定閱讀
教科書: W. K. Nicholson, Introduction to Abstract Algebra, 3rd edition, John Wiley & Sons, 2007 
評量方式
(僅供參考)
 
No.
項目
百分比
說明
1. 
作業 
20% 
Homework will be collected every Thursday (tutorial class). No Late Homework!  
2. 
平時考 
20% 
(Thursday tutorial class) 20 minutes for each test, total of 7-10 tests for this semester 
3. 
期中考 
30% 
the Week of November 4  
4. 
期末考 
30% 
the Week of January 7 
 
課程進度
週次
日期
單元主題
第1週
9/10,9/13  * Definition and example of groups, rings, and fields



* Division Algorithm, Euclidean Algorithm

*equivalence relation

*Integers modulo n


 
第2週
9/17,9/20  Chinese remainder theorem, Fermat's theorem, an application to crytography, permutations, binary operations 
第3週
9/24,9/27  definition of groups, basic properties of groups, classification of groups with 2,3,or 4 elements,
subgroups  
第4週
10/01,10/04  examples of subgroups, cyclic groups, order of an element, fundamental theorem of finite cyclic groups, examples of group homomorphism and group isomorphism 
第5週
10/08,10/11  basic properties of homomorphism and isomorphism, automorphism, inner automorphism, Cayley's theorem, cosets, Langrange's theorem 
第6週
10/15,10/18  Application of Lagrange's theorem, Dihedral groups, classification of groups of order 6, Euler's theorem, normal subgroups 
第7週
10/22,10/25  classification of order 8 abelian groups, definition of factor groups (quotient groups) and basic properties, commutator subgroups (derived groups), alternation groups (A_n) 
第8週
10/29,11/01  kernel, image, isomorphism theorem, homomorphic images of a group, alternating groups (A_n) 
第9週
11/05,11/08  simple groups, examples of simple groups, more on alternating groups, review for midterm exam 
第10週
11/12,11/15  definition of rings, examples, basic properties, characteristic of a ring, subrings, units, idempotents, nilpotents, division rings, fields 
第11週
11/19,11/22  integral domains, fields, field of quotients, ideals, factor rings 
第12週
11/26,11/29  prime ideals, maximal ideals, ring homomorphisms, isomorphism theorem 
第13週
12/03,12/06  applications of the isomorphism theorem, decomposition of rings, Chinese remainder theorem, polynomial rings, division algorithm of division ring, evaluation theorem, remainder theorem  
第14週
12/10,12/13  factorization of polynomial rings over fields, Gauss lemma, Eisenstein criterion 
第15週
12/17,12/20  factor rings of polynomials over a field, Kronecker's theorem, algebraic extension of a field, minimal polynomial of an algebraic element over a field 
第16週
12/24,12/27  splitting fields, finite fields 
第17週
12/31,1/03  geometric construction(ruler and compass), fundamental theorem of algebra